Hua's identity
In algebra, Hua's identity[1] named after Hua Luogeng, states that for any elements a, b in a division ring, [math]\displaystyle{ a - \left(a^{-1} + \left(b^{-1} - a\right)^{-1}\right)^{-1} = aba }[/math] whenever [math]\displaystyle{ ab \ne 0, 1 }[/math]. Replacing [math]\displaystyle{ b }[/math] with [math]\displaystyle{ -b^{-1} }[/math] gives another equivalent form of the identity: [math]\displaystyle{ \left(a + ab^{-1}a\right)^{-1} + (a + b)^{-1} = a^{-1}. }[/math]
Hua's theorem
The identity is used in a proof of Hua's theorem,[2][3] which states that if [math]\displaystyle{ \sigma }[/math] is a function between division rings satisfying [math]\displaystyle{ \sigma(a + b) = \sigma(a) + \sigma(b), \quad \sigma(1) = 1, \quad \sigma(a^{-1}) = \sigma(a)^{-1}, }[/math] then [math]\displaystyle{ \sigma }[/math] is a homomorphism or an antihomomorphism. This theorem is connected to the fundamental theorem of projective geometry.
Proof of the identity
One has [math]\displaystyle{ (a - aba)\left(a^{-1} + \left(b^{-1} - a\right)^{-1}\right) = 1 - ab + ab\left(b^{-1} - a\right)\left(b^{-1} - a\right)^{-1} = 1. }[/math]
The proof is valid in any ring as long as [math]\displaystyle{ a, b, ab - 1 }[/math] are units.[4]
References
- ↑ Cohn 2003, §9.1
- ↑ Cohn 2003, Theorem 9.1.3
- ↑ "Is this map of domains a Jordan homomorphism?". https://math.stackexchange.com/q/161301.
- ↑ Jacobson 2009, § 2.2. Exercise 9.
- Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6.
- Jacobson, Nathan (2009). Basic algebra. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-47189-1. OCLC 294885194.
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